Di erential Geometric Description of 3D Scalar

Dierential Geometric Description of 3D Scalar Images Alfons H. Salden Bart M. ter Haar Romeny Max A. Viergever Luc M. J. Florack Jan J. Koenderink 3D Computer Vision, Utrecht University Hospital, Room E.01.334, Heidelberglaan 100, 3584 CX Utrecht, The Netherlands, email: [email protected] 

1992 Abstract

We dene the equivalence problem with respect to the local dierential geometric structure of 3D scalar images by requiring invariance under the orthogonal group of spatial transformations and under the group of general intensity transformations. The solution of this problem is a complete set of dierential geometric features at every point and xed scale. We obtain this set by taking the local directional derivatives along the unit tangent vector of the owline and the unit principal vectors of the geodesics on the isophote of their corresponding fundamental intrinsic properties. This method of object description enables us to extract special pointsets and to determine topological and projective invariances.

1 Introduction The problem of describing an image may be cast in an equivalence problem: we have to specify the group of transformations of dependent and independent variables under which the properties of an image remain the same. For 2D scalar images we have already determined a complete set of local dierential geometric features invariant under both the orthogonal group of spatial transformations and the group of general intensity transformations [2]. This set corresponds to the intrinsic dierential geometric properties of isophotes and owlines. Using scale space theory [5] we may compute all manifest invariant local features of those curves on all scales. The aim of the present paper is to establish a complete description of 3D scalar images by considering a similar equivalence problem as mentioned above. In [9] and [10] the author determines the fundamental intrinsic properties of  This work is supported by the Netherlands Organisation of Scientic Research, grant nr. 910-408-09-1

1

parametrized curves and implicitly dened surfaces of 3D scalar images, but does not x a local gauge in order to express higher order dierential geometric features of those objects in manifest invariant form. Deriving such a local gauge in a manifest invariant form we solve the equivalence problem for objects that are at a xed scale invariant under the orthogonal group of spatial transformations and the group of general intensity transformations. The only objects invariant under both groups of transformations are owlines and isophotes [2]. These geometric objects are described by a complete set of local dierential geometric features, which we obtain by taking the derivatives with respect to the gauge coordinates of their fundamental intrinsic properties.

2 Local Dierential Geometry of 3D Scalar Images We dene, using scale space theory [5], a 3D scalar image to be a spatial intensity distribution L:

L : <  > :

1

2 3

with

= (s1 + s2 ) + l31 p = ? 21 (s1 + s2 ) + l31 + ip2 3 (s1 ? s2 ) = ? 21 (s1 + s2 ) + l31 ? i 2 3 (s1 ? s2 )

s

1

s2

= [r + (q3 + r2 ) 2 ] 3 1 1 = [r ? (q3 + r2 ) 2 ] 3 1 1

11

(67)

(68)

and

(

q = r =

l2

? l1 2

3 (3 3

9

l ?l1 l2 ) + l31 6

(69)

27

Thus we arrive at the pure second order algebraic classication of table (1) and may distinguish concave elliptic, convex elliptic, one and two sheet hyperbolic generic voxels. These voxels dene isophotes and owlines with respect to the determinant of the Hessian, which may be characterized by their intrinsic properties. Similarily we may of course investigate higher xed and mixed order algebraic dierential structure. Considering a discretized parabolic surface it becomes now very easy to extract its dierential geometric and topological properties. (i ; j ; k ) < < <